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Reference
Energy relaxation at quantum Hall edge
LEVKIVSKYI, Ivan, SUKHORUKOV, Eugene
Abstract
In this work, we address the recent experiments [C. Altimiras et al., Nat. Phys. 6, 34 (2010); H.
le Sueur et al., Phys. Rev. Lett. 105, 056803 (2010);C. Altimiras et al., Phys. Rev. Lett. 105, 226804 (2010)], where an electron distribution function at the quantum Hall (QH) edge at filling factor ν=2 has been measured with high precision. It has been reported that the energy of electrons injected into one of the two chiral edge channels with the help of a quantum point contact (QPC) is equally distributed between them, in agreement with earlier predictions, one being based on the Fermi gas approach [A. M. Lunde et al., Phys. Rev. B 81, 041311(R) (2010)] and the other utilizing the Luttinger-liquid theory [P. Degiovanni et al., Phys. Rev. B 81, 121302(R) (2010)]. We argue that the physics of the energy relaxation process at the QH edge may in fact be more rich, providing the possibility for discriminating between two physical pictures in experiment. Namely, using the recently proposed nonequilibrium bosonization technique [I. P. Levkivskyi et al., Phys. Rev. Lett., 103, 036801 (2009)], we evaluate the electron [...]
LEVKIVSKYI, Ivan, SUKHORUKOV, Eugene. Energy relaxation at quantum Hall edge. Physical Review. B, Condensed Matter , 2012, vol. 85, no. 7
DOI : 10.1103/PhysRevB.85.075309
Available at:
http://archive-ouverte.unige.ch/unige:36325
Disclaimer: layout of this document may differ from the published version.
Energy relaxation at quantum Hall edge
Ivan P. Levkivskyi and Eugene V. Sukhorukov
D´epartement de Physique Th´eorique, Universit´e de Gen`eve, CH-1211 Gen`eve 4, Switzerland (Received 15 November 2011; published 10 February 2012)
In this work, we address the recent experiments [C. Altimiraset al.,Nat. Phys.6, 34 (2010); H. le Sueur et al.,Phys. Rev. Lett.105, 056803 (2010);C. Altimiraset al.,Phys. Rev. Lett.105, 226804 (2010)], where an electron distribution function at the quantum Hall (QH) edge at filling factorν=2 has been measured with high precision. It has been reported that the energy of electrons injected into one of the two chiral edge channels with the help of a quantum point contact (QPC) is equally distributed between them, in agreement with earlier predictions, one being based on the Fermi gas approach [A. M. Lundeet al.,Phys. Rev. B81, 041311(R) (2010)]
and the other utilizing the Luttinger-liquid theory [P. Degiovanniet al.,Phys. Rev. B81, 121302(R) (2010)]. We argue that the physics of the energy relaxation process at the QH edge may in fact be more rich, providing the possibility for discriminating between two physical pictures in experiment. Namely, using the recently proposed nonequilibrium bosonization technique [I. P. Levkivskyiet al.,Phys. Rev. Lett.103, 036801 (2009)], we evaluate the electron distribution function and find that the initial “double-step” distribution created at a QPC evolves through several intermediate asymptotics before reaching eventual equilibrium state. At short distances, the distribution function is found to be asymmetric due to non-Gaussian current noise effects. At larger distances, where noise becomes Gaussian, the distribution function acquires symmetric Lorentzian shape. Importantly, in the regime of low QPC transparenciesT, the width of the Lorentzian scales linearly withT, in contrast to the case of equilibrium Fermi distribution, the width of which scales as√
T. Therefore, we propose to do measurements at low QPC transparencies. We suggest that the missing energy paradox may be explained by the nonlinearities in the spectrum of edge states.
DOI:10.1103/PhysRevB.85.075309 PACS number(s): 73.23.−b, 03.65.Yz, 85.35.Ds
I. INTRODUCTION
A two-dimensional electron gas (2DEG) in strong perpen- dicular magnetic field exhibits the regime of quantum Hall (QH) effect.1One of the peculiar phenomena specific to this regime is the appearance of one-dimensional (1D)chiraledge states, which are quantum analogs of skipping orbits. Recent extensive experimental studies2–6of these states have led to the emergence of a new field in condensed matter physics dubbed the electron optics. On the theoretical side, there are two main points of view on the physics of QH edge states. One group of theories7suggests that at integer values of the Landau-level filling factor, the edge excitations are free chiralfermions. The second group of theories is based on the concept of the edge magnetoplasmon picture.8 The fundamental edge excitations in these theories are the charged and neutral collectiveboson modes.
The domain where these two approaches meet each other is thelow-energyeffective theory.9 In the framework of this theory, both fermion and boson excitations are two forms of the same entity. Namely, they can be equivalently rewritten in terms of each other:
ψ(x,t)∼exp[iφ(x,t)],
where ψ(x,t) is the fermion field and φ(x,t) is the boson field. However, this transformation is highly nonlinear, and in the presence of strong Coulomb interaction, fermions are not stable and decay into the boson modes, which are the eigenstates of the edge Hamiltonian.
Results of tunneling spectroscopy experiments10 reason- ably agree with the free-electron description of edge states at integer filling factors. However, the first experiment on Aharonov-Bohm (AB) oscillations of a current through the
electronic Mach-Zehnder (MZ) interferometer2has shown that the phase coherence of edge states is strongly suppressed at energies, which are inversely proportional to the interferom- eter’s size. Moreover, several subsequent experiments on MZ interferometers at filling factor ν=2 have shown puzzling results on finite bias dephasing3–6 theoretically studied in Refs.11–15. Namely, the visibility of AB oscillations in these experiments is found to have a lobe-type pattern as a function of the applied voltage bias. Such results are difficult to explain in terms of the fermion picture, while they all follow naturally from the plasmon physics,13 where the Coulomb interaction plays a crucial role. Thus, the boson picture of edge excitations might be more appropriate.
In contrast to the above-mentioned nonlocal experiments, some local measurements seem to be not able to differentiate between two physical pictures of edge states. For example, both theories predict Ohmic behavior of the tunneling current, unless it is renormalized by a nonlinear dispersion of plasmons.
Moreover, the equilibrium distribution of the bosons is equiv- alent to that of fermions (see the demonstration of this fact in Sec.IV C). Therefore, it might be interesting to investigate nonequilibrium local properties of edge states.
Nonequilibrium behavior of 1D systems has been a subject of intensive theoretical16 and experimental17 studies for a long time. However, only recently has it become possible to measure an electron distribution at quantum Hall edgef() as a function of energywith high precision.18The main idea of the experimental technique is to restore the functionf() by measuring the differential conductanceGof tunneling between two edges through a single level in a quantum dot:
G()∝∂f()/∂, (1)
FIG. 1. (Color online) Schematics of the experiments (Refs.18 and19). The shaded region is filled by the 2D electron gas in the regime of the quantum Hall effect. At filling factorν=2, there are two chiral edge states shown by the blue (the outer channel) and the black (the inner channel) lines. The QPC of the transparencyT and biased with the voltage differenceμinjects electrons into the outer channel, and thus creates a nonequilibrium electron distribution.
After the propagation along the QH edge, the distribution is detected at distancesLfrom the source with the help of a quantum dot with a single level controlled by the gate voltageVg.
where is the energy of the quantum-dot level, controlled by the gate voltage Vg. This technique has been used in experiments19 in order to investigate the energy relaxation at QH edge states at filling factorν=2. The schematics of these experiments is shown in Fig.1. The main result is that the electron distribution relaxes toward local equilibrium Fermi distribution, and the energy splits equally between the two edge channels.
The first theoretical models, based on the fermion picture20 and on the plasmon approach,21 have come qualitatively to identicalconclusions. Namely, both works predict equal distri- bution of the energy between the edge channels, in agreement with the experimental findings. In other words, based on the results of Refs.20and21alone, the experimentalists are not able19to discriminate between two alternative descriptions of the physics of QH edge. Thus, it seems to be important and timely to reanalyze the problem of the energy relaxation at the QH edge in order to make new, model-specific and distinct predictions that can be verified experimentally. This is exactly the purpose of this work.
Here, we show that the Coulomb interaction strongly affects the spectrum of collective edge excitations and leads to the formation of charged and dipole plasmons modes, which prop- agate with different velocities.22 They carry away the energy of electrons injected through the QPC and equally distribute it between edge channels at distancesLexfrom the QPC. In addition to this observation, which agrees with findings of previous works,20,21we stress that the same process splits the wave packets of injected electrons and leads to strong coupling of electrons to the noise of the QPC current. The regime of weak injection, i.e., when the transparency of the QPC is smallT 1, deserves a special consideration. In this regime, the current noise at relevant time scales becomes Markovian and, as a result, the function−∂f()/∂acquires a Lorentzian shape. (This effect resembles a well-known phenomenon of the
homogeneous level broadening.) Interestingly, the width of the Lorentzian scales asT μat smallT, whereμis the voltage bias applied to a QPC. In contrast, the width of the eventual equilibrium Fermi distribution of thermalized electrons scales as√
T μ. If thermalization takes place at longer distances LeqLex, then the intermediate regime described here may be observed in experiment with a weak injection. This would indicate that interactions strongly affect the physics at the edge and that the fermion picture becomesinappropriate.
In order to theoretically describe the experiments19 and to quantitatively elaborate the physical picture, we use the nonequilibriumbosonization technique, which has been intro- duced in our previous work.23The main idea of this approach is based on the fact that in a 1D chiral system, one can find a nonequilibrium density matrix by solving equations of motion for plasmons with nontrivial boundary conditions. Then, one can rewrite an average over the nonequilibrium state of an interacting system in terms of the full counting statistics (FCS) generators24 of the current at the boundary. In the situation considered in this paper, because of chirality of QH edge states, interactions do not affect the transport through the QPC alone.
This leads to a great simplification because, in the Markovian limit, the FCS generator for free electrons is known.24
The structure of the paper is following: In Sec. II, we describe the nonequilibrium bosonization technique in some details. Next, we use this technique in Sec. III in order to find the electron correlation function for different distances from the QPC. Finally, we use these results to find the electron distribution function in Sec.IV, and present our conclusions in Sec.VI. Several important technical steps and the phenomena resulting from the nonlinearity of the spectrum of plasmons are described in Sec.Vand the Appendices.
II. NONEQUILIBRIUM BOSONIZATION
We note that the relevant energy scales (voltage bias, temperature, etc.) in recent mesoscopic experiments with the QH edge states3–6,18are much smaller than the Fermi energy.
Therefore, it is appropriate to use the low-energy effective theory9of the QH edge. One of the advantages of this theory is that it allows us to take into account strong Coulomb interactions in a straightforward way.13However, an additional complication arises from the fact that in experiments,18,19the injection into one of the two edge channels creates a strongly nonequilibrium state. We, therefore, start by recalling in this section the method of nonequilibrium bosonization, proposed earlier in Ref.23, which is suitable for solving the type of a problem that we face. Throughout the paper, we sete=h¯ =1.
A. Fields and Hamiltonian
According to the effective theory of QH edge,9 the collective fluctuations of the charge densitiesρα(x) of the two edge channels,α=1,2, at filling factor ν=2 are the only relevant degrees of freedom at low energies. These charge densities may be expressed in terms of thechiralboson fields φα(x),
ρα(x)=(1/2π)∂xφα(x), (2)
which satisfy the following commutation relations:
[φα(x),φβ(y)]=iπ δαβsgn(x−y). (3) The vertex operator
ψα(x)= 1
√aeiφα(x) (4) annihilates an electron at pointx in the edge channelα. The constant a in the prefactor is the ultraviolet cutoff, which is not universal and will be omitted and replaced by other normalizations. One can easily check, with the help of the commutation relations (3), that the operators (4) indeed create a local charge of the value 1 at pointx, and satisfy fermionic commutation relations.
Close to the Fermi level, the spectrum of electrons may be linearized; therefore, the free-fermion partH0of the total QH edge HamiltonianH=H0+Hinttakes the following form:
H0= −ivF
α
dx ψα†∂xψα, (5) where the bare Fermi velocityvF is assumed to be the same for electrons at both edge channels. The second contribution to the edge Hamiltonian describes the density-density Coulomb interaction
Hint=(1/2)
α,β
dx dy Uαβ(x−y)ρα(x)ρβ(y), (6) which is assumed to be screened at distancesd smaller than the characteristic length scale L in experiments,3–6,18,19 i.e., Ld. Therefore, we may write
Uαβ(x−y)=Uαβδ(x−y). (7) Screening may occur due to the presence of either a back gate or several top gates. We show in the following that the assumption (7) results in the linear spectrum of charge excitations. This approximation seems to be reasonable, agrees well with some experimental observations such as an Ohmic behavior of the QPC conductance at low voltage bias, and eventually does not strongly affect our main results.
Nevertheless, we relax this assumption and investigate the effects of weak and strong nonlinearities in the spectrum of charge excitations.
After taking into account the relations (2) and (4) and applying the point-splitting procedure, we arrive at the edge Hamiltonian of the quadratic form in boson fields
H= 1 8π2
α,β
Vαβ
dx ∂xφα(x)∂xφβ(x), (8) which nevertheless contains free-fermion contribution as well as the Coulomb interaction potential
Vαβ=2π vFδαβ+Uαβ. (9) Equations (3), (4), (8), and (9) complete the description of the QH edge at low energies.
The experimentally found18,19electron distribution function at the outmost QH edge channel is given by the expression
f()=
dt e−itψ1†(L,t)ψ1(L,0). (10)
By rewriting this expression via the boson fields, we finally obtain
f()=
dt e−itK(t), (11a) K(t)= e−iφ1(L,t)eiφ1(L,0). (11b) where we have introduced the electron correlation functionK, evaluated at coincident points at distanceLfrom the QPC. The proportionality coefficient in Eq. (11b) may be corrected later from the condition thatf() takes a value 1 for energies well below the Fermi level (see, however, the discussion in Sec.V for further details). In equilibrium, in order to evaluate the correlation function on the right-hand side of this equation, one may now follow a standard procedure25of imposing periodic boundary conditions on the boson fields and diagonalizing the Hamiltonian (8). In our case, however, the average in (11) has to be taken over a nonequilibrium state created by a QPC. Attempting to express such a state entirely in terms of bosonic degrees of freedom is a complicated, and not a best, way to proceed. We circumvent this difficulty by applying a nonequilibrium bosonization technique proposed in our earlier work.23 This technique is outlined in the following in some detail.
B. Equations of motion, boundary conditions, and FCS The Hamiltonian (8), together with the commutation relations (3), generates equations of motion for the fieldsφα, which have to be complemented with boundary conditions26
∂tφα(x,t)= − 1 2π
β
Vαβ∂xφβ(x,t), (12a)
∂tφα(0,t)= −2πjα(t). (12b) The last equation follows from the charge continuity condition ∂tρα+∂xjα=0 and the definition (2). Thus, the operatorjα(t) describes a current through the boundaryx =0 in the channelα. For convenience, we place a QPC in the outer channelα=1 right before the boundary, so that the operator j1(t) describes an outgoing QPC’s current.
The key idea of the nonequilibrium bosonization approach is to replace the average in Eq. (11) by the average over temporal fluctuations of currentsjα, the statistics of which is assumed to be known. Indeed, although in general the fieldsφα
influence fluctuations of the currentsjα, leading to such effects as the dynamical Coulomb blockade27and cascade corrections to noise,28in the case of chiral fields describing QH edge states, no back-action effects arise.11,13As a consequence, at integer filling factors, the electron transport through a single QPC is not affected by interactions, which seems to be an experimental fact.6,18 Therefore, by solving Eqs. (12), one may express the correlation functions of the fieldsφαin terms of the generator of FCS (Ref.24):
χα(λ,t)= eiλQα(t)e−iλQα(0). (13) Here, averaging is taken overfreeelectrons, and the operators
Qα(t)= t
−∞
dtjα(t) (14)
may be viewed as a total charge in the channelαto the right of the boundary atx=0.
To prove the connection of the electron correlations in (11) to the generating functions (13), we come back to the discussion of the interaction effects, which are in fact encoded in a solution of the equations of motion (12a). The long-range character of the Coulomb interaction leads to the logarithmic dispersion in the spectrum of collective charge excitations, the physical consequences of which are discussed in Secs.IV andV. For a moment, to simplify equations (12a), we have assumed screening of the Coulomb potential at distances d shorter than the characteristic length scalevF/μ, which is of the order of few microns in recent experiments. Nevertheless, it is very natural to assume that the screening lengthdis much larger than the distanceabetween edge channelsd a, which does not exceed a few hundred nanometers. Therefore, one can write
Uαβ=π u, u/vF ∼ln(d/a)1, (15) i.e., the in-channel interaction strength is approximately equal to the intrachannel. As a result, the spectrum of collective charge excitations splits into two modes: a fast charged mode φ˜1with the speeduand a slow dipole mode ˜φ2with the speed vvF.
It is important to stress that the conditiond a, leading to (15), results in a sort of universality: the solution of equations of motion (12a) in terms of the charged and dipole mode
φ1(x,t)= 1
√2[ ˜φ1(x−ut)+φ˜2(x−vt)], (16a) φ2(x,t)= 1
√2[ ˜φ1(x−ut)−φ˜2(x−vt)] (16b) is only weakly sensitive to perturbations of our model, in particular to those that account for different bare Fermi velocities of edge channels and slightly different interaction strengths.
By applying now boundary conditions (12b) to the result (16), we finally solve equations of motion in terms of the boundary currents:
φ1(x,t)= −π tu
−∞dt[j1(t)+j2(t)]
−π tv
−∞dt[j1(t)−j2(t)], (17a) φ2(x,t)= −π
tu
−∞
dt[j1(t)+j2(t)]
+π tv
−∞
dt[j1(t)−j2(t)], (17b) where we have introduced notations
tu=t−x/u, tv=t−x/v. (18) Finally, using the definition (14), we arrive at the solution in the compact form
φ1(x,t)= −π[Q1(tu)+Q2(tu)+Q1(tv)−Q2(tv)], (19) and to a similar expression for the inner channel. The physical meaning of this result is rather simple: when charges are injected into the channel α=1 and 2, they excite charged
FIG. 2. (Color online) Schematic illustration of the Coulomb interaction effect at the QH edge at filling factorν=2. The electron wave packet of the chargeecreated in the outer edge channel (black, lower line) decays into two eigenmodes of the Hamiltonian (8), the charged and dipole mode, which propagate with different speeds and carry the chargee/2 in the outer channel. As a result, the wave packets do not overlap at distances larger than their width, and contribute independently to the electron correlation function with the coupling constantλ=π(Ref.29). A similar situation arises when an electron is injected in the inner channel (blue, upper line), however, in this case the charged and dipole states carry opposite charges at the outer channel. Thus, there are four independent contributions to the correlation function in the outer edge channel.
and dipole modes (note the minus sign in the fourth term on the right-hand side), which have different propagation speeds uandv. As a result, these charges arrive at the observation pointx with different time delaysx/uandx/v, and make a contribution to the fieldφ1at different times (18).
When substituting this result into the correlation function in Eq. (11b), one may use the statistical independence of the current fluctuations at different channels and split the exponential functions accordingly:
K(t)=
eiπ[Q1(tu)+Q1(tv)]e−iπ[Q1(tu−t)+Q1(tv−t)]
×
eiπ[Q2(tu)−Q2(tv)]e−iπ[Q2(tu−t)−Q2(tv−t)]
. (20) In the rest of the paper, we will be interested in the correlation function at relatively long distances Lvτc, where τc 1/μ is the correlation time of fluctuations of the current through a QPC. (We show below that at this length scale, the energy exchange between two channels takes place.) In this case, the partitioned chargesQα, taken at different times tu andtv, are approximately not correlated, as illustrated in Fig.2. This assumption is quite intuitive and may be easily checked using Gaussian approximation. We, finally, arrive at the following important result:
K(t)=χ12(π,t)χ2(−π,t)χ2(π,t), (21) i.e., the electronic correlation function (20) may indeed be expressed in terms of the FCS generator (13).
III. ELECTRON CORRELATION FUNCTION The expression (21) presents formally a full solution of the problem of evaluation of an electron correlation function.
Generators of the FCS for free electrons in this expression, de- fined as (13), may be represented as a determinant of a single- particle operator,24and eventually evaluated, e.g., numerically.
However, a further analytical progress is possible in a number of situations, which are important for understanding physics of the energy relaxation processes. In particular, we show in this section that for the case of equilibrium fluctuations of the boundary currents, the correlation function (21) as well as
the electron distribution function (11) acquire an equilibrium free-fermionic form. The electron correlation function may also be found analytically away from equilibrium for the case of a Gaussian noise. Interestingly, in the short-time limit t 1/μ, the main contribution to the correlation function comes from zero-point fluctuations of boundary currents, and it behaves as a free-fermion correlator, i.e., it scales as 1/t. In the long-time limitt 1/μ, the nonequilibrium zero-frequency noise dominates, and the electron correlation function decays exponentially with time. This is exactly the limit where a non-Gaussian Markovian noise should also be taken into account.
A. Gaussian noise
In the context of the noise detection physics,24,30 the dimensionless counting variableλin the expression (21) for the FCS generator plays the role of a coupling constant.
Typically, it is small, λ1, so that the contributions of high-order cumulants of current noise to the detector signal are negligible.30 In contrast, in the physical situation that we consider in this paper,λ= ±π, implying that the shape of the distribution function may be strongly affected by high-order current cumulants. Nevertheless, it is instructive to first consider Gaussian fluctuations, simply truncating the cumulant expansion at second order in λ. In this case, the correlation function (20) may be evaluated exactly. There are many reasons for starting the analysis from considering an example of a Gaussian noise: First of all, in equilibrium, the current fluctuations in a chiral 1D system are always Gaussian. Second, as we show in AppendixC, the dispersion of the charged and dipole modes leads to a suppression of higher-order cumulants at large distances L. Finally, on the Gaussian level, it is easier to investigate and compare contributions of zero-point fluctuations and of nonequilibrium noise to the electron correlation function.
Thus, by expanding the logarithm of the right-hand side of Eq. (13) to second order inλand accounting for Eq. (14), we obtain
ln[χα(λ,t)]=iλjαt−λ2Jα(t). (22) Here, the Gaussian contribution of current fluctuations δjα(t)≡jα(t)− jαis given by the following integral:
Jα(t)= 1 2π
dω Sα(ω)
ω2+η2 (1−e−iωt), η→0 (23) where the nonsymmetrized noise power spectrum is defined as
Sα(ω)=
dt eiωtδjα(t)δjα(0). (24) In what follows, we apply this result for the evaluation of the electron correlation function in the case of equilibrium boundary conditions and in the case of a Gaussian noise far away from equilibrium.
1. Equilibrium boundary conditions
One may propose the following simple test of the nonequi- librium bosonization method: Let us consider an infinite QH edge. In equilibrium, the charge densities and edge currents
exhibit thermal fluctuations. This is the case, in particular, for the currentsjα through the cross section x=0, which are considered to be boundary conditions for the fieldφα in our theory. Therefore, one may evaluate the electron correlation function using these boundary conditions and compare it with the result of the standard equilibrium bosonization technique,25applied to a chiral 1D system.13
In equilibrium,jα =0. The current noise power spectrum is given by the fluctuation-dissipation relation31
Sα(ω)≡
dt eiωtjα(t)jα(0) = 1 2π
ω
1−e−βω. (25) By substituting this expression into Eq. (23), one obtains
ln[χα(λ,t)]= − λ2 4π2
dω ω
1−e−iωt
1−e−βω . (26) This integral may be evaluated by expanding the integrand in Boltzmann factors e±βω and integrating each term. By substituting the result (forλ=π) into Eq. (21), we arrive at the following expression for the electron correlation function in the case of equilibrium boundary conditions:
K(t)∝ β−1
sinh(π t /β), (27)
which is, in fact, the equilibrium fermionic correlation func- tion. The straightforward calculations of the integral (11a) give, naturally, the equilibrium distribution functionf1()= 1/(1+eβ)≡fF(), where we have fixed the normalization constant, as explained above. Thus, for chiral, interacting quasi-1D systems with linear spectrum equilibrium bosons also implies equilibrium distribution of fermions.
It is instructive to compare this result with the known expression for the electron correlation function atν=2, found earlier in Ref.13with the help of the standard bosonization technique
K(t)=β−1
sinh
x−y−vt vβ/π
sinh
x−y−ut uβ/π
−1/2
.
(28) Forx =y, details of the interaction leading to wave-packet splitting (see Fig.2) vanish, and one obtains the expression (27), thus validating our approach. Moreover, the free- fermionic character of the correlation function at coincident points (27) justifies the assumption underlying the nonequi- librium bosonization procedure that the FCS generators (13) may be taken as for free electrons.
2. Gaussian noise away from equilibrium
For a QPC far away from equilibrium, βμ1, one may simply set the temperature to zero. Straightforward calculations based on the scattering theory32give the following result for the spectral density of noise (24) of a QPC:
Sα(ω)=Sq(ω)+RαTαSn(ω), (29) where Tα =1−Rα is the transparency of a QPC (i.e., the average occupation in the channelα),Sq(ω)=(1/2π)ωθ(ω) is the quantum, ground-state spectral function, andSn(ω)=
±Sq(ω±μ)−2Sq(ω), is the nonequilibrium contribu-
FIG. 3. (Color online) Two contributions to the spectral density of noise (29). Left panel: Quantum contributionSq(ω) generated by the incoming Fermi sea. This contribution vanishes at low frequencies Sq(0)=0, but dominates the behavior of the correlator (23) at short times t μ1. Right panel: Nonequilibrium contribution Sn(ω) dominates at long timest μ1, i.e., in the Markovian limit.
tion (see Fig.3). Note that the noise power (29) differs from that for a nonchiral case.30
Evaluating the integral (23), we arrive at the following expression:
Jα(t)=(1/4π2)[lnt+2RαTαf(μt)], (30) where the logarithm of time originates from the quantum contribution Sq, and the dimensionless function f(μt), describing nonequilibrium noise, is given by the integral
f(μt)= 1
0
ds1−s
s2 [1−cos(μt s)]. (31) This function has a quadratic behaviorf(μt)(μt)2/4 at short timesμt 1, while in the long-time (Markovian) limitμt 1, the dominant contribution to this function is linear in time:f(μt)(π/2)|μt|.
For the purpose of further analysis, we need the electron correlation function in the long-time limit. Taking into account that jα =μTα/2π, we find the cumulant generating function
ln[χα]= iλ 2πμTαt
− λ
2π 2
(lnt−π RαTα|μt|), μt 1. (32) Finally, substituting this result into Eq. (21), and settingT1= T andT2=1 according to the situation shown in Fig.1, we obtain the electron correlation function in the long-time limit:
K(t)∝t−1eiμT t−π RT μ|t|/2, μt1. (33) Note that the expression (33) contains the quantum contribu- tion in the form of a single pole, as for free fermions, as well as the nonequilibrium contribution in the form of an exponential envelope, the width of which is determined by the noise power at zero frequency S1(0)=RT μ/2π. The phase shift of the correlator is determined by the “average” voltage bias μ =μT of the incoming stream of electrons, diluted by the QPC. In the next section, we show that this mean-field-like effect of the dilution is strongly modified by a non-Gaussian component of noise.
B. Non-Gaussian Markovian noise
Here, we consider non-Gaussian noise and show that the contribution of high-order cumulants of current to the
correlation function is not small. Note that the quantum ground-state part of the current noise Sq, which dominates at short times, is pure Gaussian. Therefore, the denominator in the expression (33) remains unchanged. In the long-time, Markovian limit, the dominant contribution to the FCS gener- ator comes from the nonequilibrium part of noise, which, e.g., is described bySnin Gaussian case. For a QPC, the Markovian FCS generator is given by the well-known expression24 for a binomial process
χ1(λ,t)=(R+T eiλ)N, (34) where N =μt /2π is the total number of electrons that contribute to noise. By applying the analytical continuation λ→π, we obtain
ln[χ1(π,t)]=μt
2π [ln|T −R| +iπ θ(T −R)]. (35) By substituting this expression to the correlation function (21), we arrive at the result
K(t)∝t−1eiθ(T−R)μt+ln|T−R|μ|t|/π, (36) where the imaginary part of the exponent determines the effective voltage bias, while the real part is responsible for dephasing.
Interestingly, at the point T =1/2, the dephasing rate is divergent, and the effective voltage bias drops to zero abruptly forT <1/2. It has been predicted in Ref.23 that this behavior may lead to a phase transition in the visibility of Aharonov-Bohm oscillations in electronic Mach-Zehnder interferometers. We will argue in the following that no sharp transition arises in the electron distribution function. However, it leads to its strong asymmetry with respect to the average voltage biasμ =T μof the outer channel.
IV. ELECTRON DISTRIBUTION FUNCTION In this section, we use the results (27), (33), and (36) for the correlation function of electrons to evaluate and analyze the electronic distribution function. We start by noting that the experiments (Refs. 18 and 19) are done in a particular regime of strong partitioning T ≈0.5 at the QPC injecting current to the channelα=1. This detail, which seems to be irrelevant from the first glance, is in fact of crucial importance.
Indeed, as it follows from the expressions (33) and (36), the main contribution to the integral (11a) for the correlation function comes from timest of the order of the correlation timeτc1/μ, where our results based on the Markovian noise approximation are, strictly speaking, not valid. However, the numerical calculations show that the nonequilibrium distribution in this regime is very close to the equilibrium one. Therefore, the actual equilibration of electrons, which is reported in the experiment Ref.18to occur at distancesv/μ, may in fact take place at even longer distancesLeqv/μ due to an unknown mechanism (not considered here).
Indeed, if the chiral Luttinger-liquid model considered in our paper is valid, then neither the strong interaction between electrons of two edges taken alone nor the weak dispersion of plasmons resulting from a long-range character of Coulomb interaction may lead to the equilibration because the systems remain integrable. Thus, it seems to be reasonable to assume that the equilibration length Leq may indeed be
quite long. Therefore, in order to explore the physics of collective charge excitations at intermediate distances, we pro- pose to consider a regime of weak injection at the QPC:T 1.
First, we note that in this case our results (33) and (36) may indeed be used to evaluate the electron distribution function because the main contribution to the integral (11a) arises from Markovian time scales. Second, and more importantly, in this regime the electron distribution function acquires a strongly nonequilibrium form and the width of the order ofT μ, which plays a role of the new energy scale. Moreover, the advantage of the weak injection regime is that it allows us to investigate a nontrivial evolution of the distribution function, which arises due to bosonic, collective character of excitations and goes via several well distinguishable steps.
A. Short distances
At distances of the order of the energy exchange length scale
Lex≡v/μ, (37)
the initial double-step distribution function is strongly per- turbed by the interaction between channels. As we argued in Sec. II B, at distances LLex, the charged and dipole modes split and make independent contributions to the electron correlation function. Therefore, we may rely on the result (36).
By applying the limitT 1 to this expression and evaluating the Fourier transform, we find
−∂f()
∂ = ng/π
2+ng2 , ng =2T μ/π. (38) Here, the missing prefactor in the correlation function has been fixed by the requirement that f()=1 at → −∞.
Thus, we conclude that the energy derivative of the distribution function acquires a narrow Lorentzian peak, which is shifted with respect to the average biasμ =T μand centered at =0. The last effect is a unique signature of the non-Gaussian character of noise. Because of the electron-hole symmetry of the binomial process, in the limitR 1 the Lorentzian peak obviously has a widthng=2Rμ/π and centered at =μ.
We stress again that the result (38) holds only for small enough energies close to the Fermi level, namely, for ||<
μ, where the main contribution arises from the noise in the Markovian limit. In fact, the result (38) fails at large energies in a somewhat nontrivial way. Namely, it is easy to see that any electron distribution function has to satisfy the sum rule
μ ≡0+ ∞
0
d f()= − ∞
−∞
d ∂f()/∂, (39) where 0 is the cutoff well below the Fermi level, and the
“average” biasμ =T μin the case of linear dispersion of plasmons. This sum rule simply expresses the requirement of the conservation of the charge current and implies a certain amount of asymmetry in the distribution function. In the present case, such an asymmetry arises in the power-law tails of the function−∂f()/∂and originates from quantum nonequilibrium noise. It can be seen in Fig.4.
Moreover, at energies of the order ofμ, the power-law behavior of the function (38) has to have a cutoff because the
FIG. 4. (Color online) Energy derivative of the electron distri- bution function−∂f/∂is shown for different distancesLfrom the QPC injecting current. The transparency of the QPC is set toT =0.05 and voltage bias isμ=40μV. Black line:−∂f()/∂ for short distancesLexLLg, so that the noise is non-Gaussian (38). Red, dashed line:−∂f()/∂ for intermediate distancesLgLLeq, where the noise is Gaussian (44). Blue, dotted line: The derivative of the Fermi distribution function at the temperature given by Eq. (47).
The dashed line is a guide for the eyes at the energy equal to the effective voltage biasμ =T μ=2μV. Inset: The same distribution functions are shown in the integrated form. They are shifted vertically by 0.2 intervals for clarity.
QPC does not provide energy much larger than the voltage bias. Quantitatively, this follows from the conservation of the energy. We demonstrate in the following that for the system with linear dispersion of plasmons, the heat flux in the outer channel can be written entirely in terms of the single-electron distribution function (in unitse=h¯ =1)
Im=(1/2π)
d [f()−θ(μ −)], (40) as in the case of free electrons. We use the subscript “m” in order to emphasize the fact that it is this quantity that has been measured in the experiment (Ref. 19). In Sec. V, we show that at distancesLLexthe total heat flux injected at a QPC splits equally between two edge channels; therefore, integrating Eq. (40) by parts and substituting the heat flux for a double-step distribution, we obtain
Im= −(T μ)2
4π − 1
4π
d 2∂f()
∂ = T R(μ)2
8π (41)
forLLex. One can see from Eq. (38) that indeed the power- law behavior has to have a cutoff at || ∼μ. We stress, however, that this summation rule is less universal than the one given by Eq. (39) because it accounts only for a single-particle energy of electrons and fails in the case of a nonlinear spectrum of plasmons, considered in Sec.Vin detail.
B. Intermediate distances
So far, we have considered the case of a linear spectrum of plasmons. This is a reasonable assumption, taking into account
the fact that nonlinear corrections in the spectrum of plasmons lead to a nonlinear corrections in the Ohmic conductance of a QPC. However, the experiments (Refs.18and19) seem to be done in the Ohmic regime. Nevertheless, even in the case of a weak nonlinearity in the spectrum of the the both modes of the sort33
kj(ω)=ω/vj +γjω2sign(ω), v1=u, v2=v, (42) barely seen in the conductance of a QPC, an intermediate length scale Lg may arise at which high-order cumulants of current are suppressed, and the noise becomes effectively Gaussian. This situation occurs when the wave packets of the original widthv/(T μ) overlap. A simple estimate using the nonlinear correction (42) gives the length scale
Lg=1/γ(T μ)2, γ ≡min(γj). (43) We support this conclusion by rigorous calculations in AppendixC.
The nonlinearity in the spectrum is weak ifγ vT μ1.
This implies that LgLex, and leads to the possibility to observe non-Gaussian effects at distances LexL Lg discussed in the previous section. Obviously, the same requirement also guarantees that dispersion corrections to the Ohmic conductance of a QPC are small. This allows us to neglect corrections to the quantum part of the electron correlation function and to use the result (33) for a Gaussian noise. By substituting this result to Eq. (11a), we obtain
−∂f()
∂ = ng/π
(− μ)2+g2, g=π T μ/2 (44) in the caseLgLLeq. One can see that the width of the function (44) is almost twice as large compared to that in the function (38). Moreover, the function (44) satisfies the sum rule (39). Therefore, we do not expect any asymmetry in the high-energy tails of this function, in contrast to the situation with the non-Gaussian noise. The comparison of distribution functions in these two regimes is shown in Fig.4.
So far, we have considered a situation where both charged and dipole modes are dispersive. If for some reason the dispersion of one of the modes is negligible, then higher-order cumulants are suppressed only by a factor of 2. The derivative of the electron distribution function in this situation is given by the Lorentzian
∂f()
∂ = (ng+g)/2π
(− μ/2)2+(ng+g)2/4 (45) centered atμ/2=μT /2 with the width (ng+g)/2= (1/π+π/4)μT. This is because one mode brings only the Gaussian component of the Markovian noise, while the other one brings full non-Gaussian noise.
C. Long distances
Next, we consider the distribution function at long distances LLeqafter the equilibration takes place. The temperature of the eventual equilibrium distribution may be found from the conservation of energy. In the next section, we show that the heat flux produced at QPC splits equally between two edge
states. In the situation of linear dispersion, the distribution function acquires the form
f()= 1
1+e(−μ)/ eq. (46) The possibility of such an equilibration process is suggested by the fact that the equilibrium distribution of bosons implies the equilibrium distribution for electrons, as has been shown in Sec.III A 1. Obviously, the distribution (46) satisfies the sum rule (39), while the energy conservation condition (41) may now be used in order to find the effective temperature:
eq=
3T /2π2μ, (47)
where we have usedT 1.
We conclude that the width of the equilibrium distribution scales as √
T, in contrast to the case of a nonequilibrium distribution at shorter distances from the current source, where it scales linear inT. Therefore, ifT is small, equilibrium and nonequilibrium distributions may easily be distinguished, as illustrated in Fig.4. In the situation where the dispersion can not be neglected, the equilibrium distribution of fermions is not given by the Fermi function (46). This situation deserves a separate consideration, which is provided in the next section.
V. MEASURED AND TOTAL HEAT FLUXES We have seen that in the case of weakly dispersive plasmons γ vT μ1, the nonlinearity in the spectrum leads to the suppression of high-order cumulants of current noise at relatively long distances, which strongly affects the distribution function. On the other hand, the direct contribution of the nonlinear correction in the spectrum to local physical quantities, such as the QPC conductance and the heat flux, is small and has been so far neglected. Nevertheless, it may manifest itself experimentally in a quite remarkable way. In this section, we show that the nonlinearity in the the plasmon spectrum contributes differently to the measured heat flux (40) and to the actual heat flux expected from the simple evaluation of the Joule heat. As we demonstrate below, this may, under certain circumstances, explain the missing energy paradox in the experiment Ref.19.
We start by noting that the measured flux (40) at the distance Lfrom the QPC may be expressed entirely in terms of the excess noise spectrumSα(ω)≡Sα(ω)−(1/2π)ωθ(ω) of edge channels right after the QPC, whereSα(ω) is defined in (24).
Namely, in AppendixB, we derive the following result:
Im(L)= 1 4
∞
−∞
dω{S1(ω)[1+cos(kL)]
+S2(ω)[1−cos(kL)]}, (48) where k≡k1(ω)−k2(ω) and kj(−ω)= −kj(ω). Impor- tantly, this result holds for an arbitrary nonlinear spectrum kj(ω) of the charged and dipole modes, and for a non-Gaussian noise in general, i.e., high-order cumulants simply do not contribute.
One can easily find two important limits of Eq. (48): for L=0, we immediately obtain an expected result
Im(0)=1 2
∞
−∞
dωS1(ω), (49)
while atL→ ∞the cosine in (48) acquires fast oscillations, and we get
Im(∞)= 1 4
∞
−∞dω[S1(ω)+S2(ω)]. (50) To be more precise, this happens at LLex=v/μ. At zero temperature, S2 vanishes and the single-electron heat flux Im, created at the QPC, splits equally between edge channels:Im(∞)=Im(0)/2. Note also that in the case of linear dispersionSα=TαRαSn, whereSnis shown in Fig.3.
In the next step, we rewrite the same measured flux in terms of the plasmon distributionsnj(k)= a˜j†(k) ˜aj(k) (see AppendixB):
Im(∞)= 1 4π
j
dk
k ω2j(k)nj(k)+Iq, (51) Iq= 1
8π
j
dk k
ωj2(k)−(vjk)2
, (52) where vj =∂ωj/∂k are the plasmon speeds at k=0. The termIqis the contribution to the measured flux from quantum smearing of the zero-temperature electron distribution function f0() close to the Fermi level, which originates from a nonlinear dispersion of plasmons.
Here, an important remark is in order. The integral (52) may diverge at largekand has to be cut off at the upper limit. This is because there is no guarantee of the free-fermionic behavior of the correlatorK(t) at short times and of the zero-temperature electron distribution function f0() at large energies. Thus, the integral (40) has to be also cut off, which is what in fact is done in experiment. In contrast, the spectrum of plasmons is linear at smallk, and thus the distribution functionf0() has a free-fermionic behavior close to the Fermi level. Our definition of Iq corresponds to the normalization of f0() to have a discontinuity of the value −1 at = μ. The experimentally measuredIq may differ from the one defined in (52) by a constant, which is, on the other hand, independent on the voltage biasμ.
Next, we note that the actual total heat flux in the case of a nonlinear dispersion of plasmons acquires the completely different form34
Ih= 1 2π
j
dk∂ωj
∂k ωj(k)nj(k), (53) and thus in general Im=Ih/2, contrary to what has been assumed in the experiment Ref. 19. This may explain the missing energy paradox. Indeed, by assuming the low ω spectrum of the general form
kj =ω/vj +γjωj, j =1,2 (54) whereγjare small, and equilibration of plasmons atL→ ∞, i.e.,nj(k)=nB(ωj/ eq)=1/[exp(ωj/ eq)−1], we obtain the missing heat flux as
Im−Iq−Ih/2=
j=1,2
cjγjvjeqj+1, (55) where the constants cj =(1/4π)
dz zj+1nB(z) are of the order of 1. This result implies that experimentally, the missing
FIG. 5. (Color online) Typical spectrum of charged plasmon in the case of the Coulomb interaction screened at distancesd1/k.
This spectrum is concave, i.e.,∂ω/∂k < ω(k)/k.
heat flux may be found by investigating its bias dependence and the spectrum of plasmons.
Let us consider an example where the dispersion of charged plasmon at smallkarises from the screened Coulomb interaction8
ω/k=2[K0(ka)−K0(kd)]
=2 ln(d/a)−(1/2)(kd)2ln(2/kd), (56) whereais the high-energy cutoff,dis the distance to the gate, such askakd 1, andK0is the MacDonald function. The low-kasymptotics of this spectrum is illustrated in Fig.5. One can see that the spectrum is concave, so that in this case the measured heat flux (51) is larger than the half of the actual heat flux (53). In addition, the effect is weak, becausekd ∼0.1 in the experiment Ref.19. Thus, the dispersion of the Coulomb interaction potential alone is not able to explain the missing flux paradox. Various mechanisms of convex dispersion are still possible and will be investigated elsewhere.
VI. CONCLUSIONS
Earlier theoretical works on quantum Hall edge states at integer filling factors may be divided into two groups:
fermion-based and boson-based theories. Recent interference experiments suggest that the boson approach might be more appropriate for the description of the edge physics. However, both groups of theories give the same predictions for the local physical quantities at equilibrium. Moreover, the first theoretical works based on fermion20and boson21approaches and addressing the nonequilibrium local measurements have not been able to make qualitatively distinct predictions. In this paper, we show that it is nevertheless possible to test and differentiate between two approaches with the local nonequilibrium measurements.
We address recent experiments (Refs. 18 and 19) with quantum Hall edge states at filling factor 2, where an energy relaxation process has been investigated by creating a nonequi- librium state at the edge with the help of a QPC and reading out the electron distribution downstream using a quantum dot.
We use the nonequilibrium bosonization approach23in order to describe the gradual relaxation of initially nonequilibrium, double-step electron distribution to its equilibrium form. In the framework of this approach, the nonequilibrium initial state is encoded in the boundary conditions for the equations of mo- tions that depend on interactions. We show that the electrons excite two plasmons: fast charged and slow dipole modes.
Thus, the electron correlation function (21) is expressed in terms of the four contributions, each having the form of the
FIG. 6. (Color online) Different length scales for energy relax- ation processes and corresponding distribution functions in each regime are schematically shown. Red curve: The initial double-step distribution function. Black curve: At distancesLμ/v, the dis- tribution function is strongly asymmetric with respect to the “average”
bias μ =T μ. Green curve: At distancesL1/γ(T μ)2, the distribution function is a Lorentzian with the width that scales asT μ. Blue curve: The final equilibrium Fermi function at large distances. For small transparencies, its width scales as√
T μ.
generator of FCS of free electrons with the coupling constant λ=π. By evaluating the Fourier transform of this function, we find the electron distribution function.
Before reaching eventual equilibrium form, the distribution function evolves via several steps, where its energy derivative acquires a Lorentzian shape:
∂f()
∂ = /π
(−0)2+2, ||μ. (57) Here, the width and centering 0 take different values in different regimes. Each of the regimes, summarized below and illustrated in Fig.6, has its own dominant process:
(i) First, after tunneling through the QPC, electrons excite plasmons, which then split in two eigenmodes: one is charged fast mode with the speed u, and the other is slow dipole mode with the speedv. This process takes place at distances Lex=v/μ, whereμis the voltage bias across the QPC.
In this regime, Eq. (21) applies, which eventually leads to the the distribution (57) with the width=ng =2μT /π, centered at0 =0.
(ii) Next, a weak dispersion of plasmons, e.g., of the form k=ω/v+γ ω2sign(ω), leads to broadening of wave packets of the energy width and to their overlap. This process takes place at distances L1/γ 2. As a result, high-order cumulants of the current injected at the QPC are suppressed at distancesLLg=1/γ(T μ)2, the noise becomes Gaus- sian, and the derivative of the electron distribution function acquires the shape (57) with the width=g =π μT /2, centered at0 =μT.
(iii) A situation is possible where the dispersion of one mode, most likely of the charged mode, is much stronger than the dispersion of the second mode, i.e.,γ1γ2. In this case, the previously described regime splits in two separate regimes. First, at distancesL=1/γ1(T μ)2, the contribution of the charged mode to high-order cumulants of noise becomes suppressed, which leads to the distribution (57) with the parameters =(ng+g)/2 and 0=μT /2. Then, at longer distances L=1/γ2(T μ)2, the noise becomes fully Gaussian.
(iv) The interaction may lead to broadening of the wave packets, but they do not decay, which implies that the interaction alone does not lead to the equilibration. This means that a different, weaker process may lead to the equilibration at distancesLeqmuch longer than the above-discussed length scales. In the tunneling regime T 1, the width of the eventual equilibrium distribution scales as √
T, in contrast to the above regimes, where it scales asT. Thus, to observe the described variety of regimes, we propose to perform measurements at large voltage biases and low transparencies of the QPC utilized to inject electrons.
Finally, we suggest a possible explanation of the paradox of missing heat flux in the experiment Ref.19. So far, we have summarized the effects of weak dispersion, which lead to the appearance of intermediate length scales. We have found that in the case of a strongly nonlinear dispersion of plasmons, the measured heat flux Im in the outmost edge channel, experimentally determined with the procedure described by Eq. (40), is different from the actual heat flux per channel Ih/2, defined by Eq. (53). The screened Coulomb interaction leads to a rather weak dispersion of the charged plasmon, and the effect is of the opposite sign because the spectrum in this case is concave. Nevertheless, other mechanisms of the convex dispersion are possible. They will be considered elsewhere.
ACKNOWLEDGMENTS
We would like to thank F. Pierre for the clarification of the experimental details and P. Degiovanni for fruitful discussions.
This work has been supported by the Swiss National Science Foundation.
APPENDIX A: SOLUTION OF EQUATIONS OF MOTION In this Appendix, we solve the equations of motion (12a) with the boundary conditions (12b) in the case of a potential Uαβ(x−y) of a finite range, where the plasmon spectrum is nonlinear. For doing this, we first write the normal mode expansion for the edge boson fields as
φα(x)=ϕα+2π·παx +
k
2π
kW[aαkeikx+a†αke−ikx]. (A1) We consider zero modes to be classical variable because the commutator [πα,ϕα]=i/W vanishes in the thermodynamic limitW →0. Then, we rewrite the operatorsaαk in the new basis ˜aj k, which diagonalizes the edge Hamiltonian (8):
˜
aj k(t)=a˜j ke−iωj(k)t, (A2) wherej =1,2 andωj(k) is the dispersion of thejth mode. In the case where the in-channel interaction strength is approxi- mately equal to the intrachannel one,Uαβ(x−y)≈U(x−y), and the interaction is strong,U(k)=
dx eikxU(x)2π vF, the transformation to the new basis is simple and universal:
a1k(t)= 1
√2( ˜a1ke−iω1(k)t+a˜2ke−iω2(k)t), (A3a) a2k(t)= 1
√2( ˜a1ke−iω1(k)t−a˜2ke−iω2(k)t). (A3b)
